It's entirely a case-by-case thing. It's dependent on the stories and characters you're working with.

This.

For a finite, beginning-middle-end story being adapted, like Watchmen, for example, I think it is very important to try to stay as close as you can. Not EVERYTHING will be able to make it to the screen, but since you have a set blueprint for the story and characters, you should try your best to stay within the lines of that. That being said, some things are ok to change or delete, as long as they do not completely alter the story. Even the ending of Watchmen got a pass from me because the giant squid wasn't as important to the story as the idea behind using the giant squid/Dr Manhattan.

For comics that have been around for decades, there is much more room for change. For example, if you make an X-Men movie, you don't have to have the original five be the team, and then bring in characters in the order in which they appeared in comics. People don't want to sit through movies about the original five and Polaris before they get Wolverine. Wolverine is the selling point. Get him in there early. One way to handle this, and this was done very well by the first movie, was to have the X-Men as a fairly established team and not worry about telling their origin story because their origin story (we were born different) isn't that interesting the way Batman's or Spider-Man's is.